Problem: In $\triangle PQR$, we have $PQ = QR = 34$ and $PR = 32$.  Point $M$ is the midpoint of $\overline{QR}$.  Find $PM$.
Explanation: We start with a diagram, including median $\overline{QN}$, which is also an altitude.  Let the medians intersect at $G$, the centroid of the triangle.


[asy]
size(100);
pair P,Q,R,M,NN;
P = (0,0);
Q = (0.5,0.9);
R = (1,0);
NN = (0.5,0);
M = (Q+R)/2;
draw(rightanglemark(Q,NN,P,2.5));
draw(M--P--Q--R--P);
draw(Q--NN);
label("$P$",P,SW);
label("$R$",R,SE);
label("$Q$",Q,N);
label("$N$",NN,S);
label("$M$",M,NE);
label("$G$",(2/3)*NN+(1/3)*Q,NW);

[/asy]

We have $NP = PR/2 = 16$, so right triangle $PQN$ gives us \begin{align*}QN &= \sqrt{PQ^2 - PN^2} = \sqrt{34^2 - 16^2} \\
&= \sqrt{(34-16)(34+16)} = 30.\end{align*} (We might also have recognized that $PN/PQ = 8/17$, so $\allowbreak QN/PQ = 15/17$.)

Since $G$ is the centroid of $\triangle PQR$, we have $GN = \frac13(QN) = 10$, and right triangle $GNP$ gives us  \[GP = \sqrt{GN^2+NP^2} = \sqrt{100+256} = 2\sqrt{25 + 64} = 2\sqrt{89}.\] Finally, since $G$ is the centroid of $\triangle PQR$, we have $PM = \frac32(GP) = \boxed{3\sqrt{89}}$.